Isometry and orbit dimensions

[quasar_4]

Hi people,

I just need to verify that I understand this correctly.

For some four dimensional manifold and group of isometries:
the dimension of the isometry group is given by the number of Killing vectors, while the dimension of the orbit group is given by the number of linearly independent Killing vectors. Right?

And in terms of understanding the orbit group: I think I understand what the orbit is, it's the points that are "traced out" from the function -- so given a point on a sphere and a rotation, we'd trace out the 2-sphere (by rotating around the polar and azimuthal angles)? So the dim of the orbit group for the rotations of a point on the sphere would have to be 2?

I'm still trying to make all the connections, but still just barely learning what Killing vectors and isometries and orbits are. :tongue2:

And are the only types of orbits Riemannian and Lorentzian? Is there such a thing as a "Null" orbit, and what does this mean exactly?

Thanks,

quasar

 

[jvicens]

Hi quasar,

You are correct in your understanding of the dimension of the isometry group and the orbit group. The isometry group is the group of transformations that preserve the metric of the manifold, and its dimension is equal to the number of Killing vectors. These are vectors that satisfy the Killing equation, which describes the invariance of a metric under a certain transformation.

The orbit group, on the other hand, is the group of transformations that map a point on the manifold to all other points on its orbit. The dimension of the orbit group is given by the number of linearly independent Killing vectors, which determine the shape and size of the orbit.

In terms of your example of a point on a sphere and rotations, you are correct that the dimension of the orbit group would be 2. This is because the sphere can be rotated in two independent directions (polar and azimuthal angles) to map a point to all other points on its orbit.

In regards to your question about the types of orbits, there are indeed Riemannian and Lorentzian orbits, which correspond to positive and negative definite metrics, respectively. There is also a third type of orbit called a null orbit, which corresponds to a metric with zero eigenvalues. This means that the orbit is "flat" and has no curvature. It can be thought of as a special case of a Riemannian or Lorentzian orbit.

I hope this helps clarify your understanding of Killing vectors, isometries, and orbits. Keep exploring and learning, and you'll continue to make connections and deepen your understanding. Good luck in your studies!